On March 11, 1998 the Minor Planet Center issued an IAU circular (no. 6837) announcing that a new orbit had been computed for the asteroid 1997 XF11, and stating: ``This nominal orbit indicates that the object will pass only 0.00031 AU from the Earth on 2028 Oct. 26.73 UT! Error estimates suggest that passage within 0.002 AU is virtually certain, this figure being decidedly smaller than has been reliably predicted for generally fainter PHAs in the foreseeable future.'' During the following 24 hours a flurry of E-mail messages, hasty computations and press releases was concluded by the precovery by K.J. Lawrence on March 12 of some observations of 1997 XF11 on films taken by the PCAS survey (led by E. Helin) in 1990. With the 1990 observations in hand, all concerned reached consensus that there was no risk of impact from this asteroid in 2028. It was difficult for the research community to reach agreement on a tricky issue of Celestial Mecahnics in just a few hours, when the algorithms used by the researchers are not well known or well understood by the community. This paper has precisely the goal of providing the necessary algorithms and software, and to test them; therefore, in this Section we shall discuss how our methods perform on the problem facing the Celestial Mechanics community on the night of March 11, 1998. It is unfortunate that these algorithms were not available at that time, because they would have enabled reseachers to eliminate the possibility of impact in 2028 with greater certainty than could be done with the linear method, and more quickly than with the Monte Carlo method.